Abstract kinetic equations with accretive collision operators

We introduce and study a new class of kinetic equations, which arise in the description of nonequilibrium macroscopic dynamics of soliton gases with elastic collisions between solitons. These equations represent nonlinear integro-differential systems and have a novel structure, which we investigate by studying in detail the class of N-component ‘cold-gas’ hydrodynamic reductions. We prove that these reductions represent integrable linearly degenerate hydrodynamic type systems for arbitrary N which is a strong evidence in favour of integrability of the full kinetic equation. We derive compact explicit representations for the Riemann invariants and characteristic velocities of the hydrodynamic reductions in terms of the ‘cold-gas’ component densities and construct a number of exact solutions having special properties (quasiperiodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed light on the structure of a continuum limit for a large class of integrable systems of hydrodynamic type and are also relevant to the description of turbulent motion in conservative compressible flows.

We consider a class of kinetic equations, equipped with a single conservation law, which generate L 1 L^{1} -contractions. We discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a (possibly) degenerate parabolic equation. The limits are obtained in the “dissipative” sense, equivalent to the notion of entropy solutions for conservation laws, which permits the use of the perturbed test function method and allows for simple proofs. A general compactness framework is obtained for the diffusive scaling in L 1 L^{1} . The radiative transport equations, satisfied by the Wigner function for random acoustic waves, present such a kinetic model that is endowed with conservation of energy. The general theory is used to validate the diffusive approximation of the radiative transport equation.

The finite element method is applied to the spatial variables of multi-group neutron transport equation in the two-dimensional cylindrical (r, z) geometry. The equation is discretized using regular rectangular subregions in the (r, z) plane. The discontinuous method with bilinear or biquadratic Lagrange's interpolating polynomials as basis functions is incorporated into a computer code FEMRZ. Here, the angular fluxes are allowed to be discontinuous across the subregion boundaries. Some numerical calculations have been performed and the results indicated that, in the case of biquadratic approximation, the solutions are sufficiently accurate and numerically stable even for coarse meshes. The results are also compared with those obtained by a diamond difference S n code TWOTRAN-II. The merits of the discontinuous method are demonstrated through the numerical studies. KEYWORDS: finite element methodneutron transportcylindrical geometryLagrange's interpolating polynomialGalerkin-type schemeCrout's methodcoarse mesh rebalancefinite difference methodnumerical solutioncomparative evaluation

Convergence of composite iterative methods for finding zeros of accretive operators

We study momentum relaxation processes of a classical and a quantum Brownian particle by considering the eigenvalue problem of the collision operators in the kinetic equations. The collision operators are anti-Hermitian with an appropriate inner product defined by an integral with a weight factor given by the inverse of the equilibrium distribution function. Owing to the weight factor, the norm of a momentum distribution function is infinite, if the distribution is characterized by a temperature higher than a threshold temperature determined by the environmental temperature. Although the eigenfunction expansion of a given distribution function with an infinite norm does not converge to a function in the Hilbert space, it has a legitimate meaning as a generalized function and defines a linear functional. We introduce an H-function through the norm which directly reflect the spectral properties of the collision operators. When the norm of the momentum distribution function diverges, the H-function reduces to a simple form characterized by only the value at the accumulation point of the spectrum of the collision operator.

In non-equilibrium classical thermostatistics, the state of a system may be described by not only dynamical/thermodynamical variables but also a kinetic distribution function. This 'double structure' bears some analogy with that in quantum thermodynamics, where both dynamical variables and the Hilbert space are involved. Recently, the concept of weak invariants has repeatedly been discussed in the context of quantum thermodynamics. A weak invariant is defined in such a way that its value changes in time but its expectation value is conserved under time evolution prescribed by a kinetic equation. Here, a new aspect of a weak invariant is revealed for the classical Fokker-Planck equation as an example of classical kinetic equations. The auxiliary field formalism is applied to the construction of the action for the kinetic equation. Then, it is shown that the auxiliary field is a weak invariant and is the Noether charge. The action is invariant under the transformation generated by the weak invariant. The result may shed light on possible roles of the symmetry principle in the kinetic descriptions of non-equilibrium systems. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

The numerical range W(T) of a linear bounded operator T on a Hilbert space is the collection of complex numbers of the form (Tx,x) with x ranging over the unit vectors in the Hilbert space. In this study, firstly, the connection between numerical range of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the first investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈L(H_{n}), H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈L(H), then numerical range of the operator A is in the form W(A)= co (⋃_{n=1}^{∞}W(A_{n}) ), where co (Ω), Ω⊂C denotes the convex hull of Ω. The numerical radius w(T) of a linear bounded operator T on a Hilbert space is a number which is given by w(T)=sup {|λ| : λ∈W(T) }. In this study, secondly, the connection between numerical radius of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the second investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈L(H_{n}), H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈L(H), then numerical radius of the operator A is in the following form w(A)= sup w(A_{n}). The Crawford number c(T) of a linear bounded operator T on a Hilbert space is given by c(T)=inf {|λ| : λ∈W(T) }. In this study, thirdly, the connection between Crawford number of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the third investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈L(H_{n}), Re(A_{n})≥0 (≤0), H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈L(H), then Crawford number of the operator A is in the following form c(A)=inf c(A_{n}). Moreover, It is known that for a linear bounded accretive operator T if W(T)⊂{z ∈C:|argz| < ϕ} for any ϕ∈[0, π/ 2), then it is called a sectorial operator with vertex γ=0 and semi-angle ϕ. In this case, T ∈S_{ϕ}(H). In this study, finally, the connection between sectoriality of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the last investigation the result obtained is as follows: If for any n≥1,H_{n} is a Hilbert space, A_{n}∈L(H_{n}),A_{n}∈S_{ϕ_{n}}(H_{n}),H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n},A∈L(H), then for some ϕ∈[0,π/2), A∈S_{ϕ}(H) the necessary and sufficient condition is sup ϕ_{n}<ϕ. At the end of this study, an example including all these connections has been given.

This paper investigates a generalized fractional coupled transport system in a Hilbert space, in which such system contains Caputo-Katugampola differential terms and Caputo-Katugampola integral terms, as well as fully nonlinear coupling functions. First, we transform this system into an iterative system by using a semi-discrete method in time based on the backward Eulerian difference format (i.e., Rothe method). Second, employing Gronwall's inequality and the properties of $ m $-accretive operators, we establish the unique solvability and several priori estimates of solutions for the iterative system. Finally, we present two illustrative examples to demonstrate the obtained theoretical results.

We consider the approximation of the inverse square root of regularly accretive operators in Hilbert spaces. The approximation is of rational type and comes from the use of the Gauss–Legendre rule applied to a special integral formulation of the fractional power. We derive sharp error estimates, based on the use of the numerical range, and provide some numerical experiments. For practical purposes, the finite-dimensional case is also considered. In this setting, the convergence is shown to be of exponential type. The method is also tested for the computation of a generic fractional power.

In this paper, we prove a strong convergence theorem for finding a common solution of a general system of finite variational inequalities for finite different inverse-strongly accretive operators and solutions of fixed point problems for a nonexpansive semigroup in a Banach space based on a viscosity approximation method by using weak contraction mappings. Moreover, we can apply the above results to find the solutions of the class of k-strictly pseudocontractive mappings and apply a general system of finite variational inequalities into a Hilbert space. The results presented in this paper extend and improve the corresponding results of Ceng et al. (2008), Katchang and Kumam (2011), Wangkeeree and Preechasilp (2012), Yao et al. (2010) and many other authors. MSC:47H05, 47H10, 47J25.

On multi-group neutron transport in planar one dimensional geometry: A solution for a localized pulsed source

The kinetic equation proposed in [1,2] for describing the behavior of a system of particles in a gas flow differs from the usual Boltzmann equation with respect to the additional terms that take into account random variations of the particle velocity under the influence of the flow. As shown in [2], the collision operator and the Brownian-type operator in the starting kinetic equation describe essentially different simultaneous physical processes of change of state of the particle system: equalization of the mean kinetic energy of the particles and change of energy due to the action of the viscous forces associated with the suspending flow. Therefore the method of solving the kinetic equation used in [2], a direct generalization of the Chapman-Enskog method of solving the kinetic equation it is necessary to investigate method of solving the kinetic equation it is necessar y to investigate the relaxation processes in the system. Moreover, the relaxation of systems of the fluidized-bed type to the continuum state is also of independent interest in connection with the analysis of fast processes in the system, i.e., processes with a characteristic duration of the order of the mean free time.

This paper presents the first application of a symmetric interior-penalty discontinuous Galerkin isogeometric analysis (SIP-DG-IGA) spatial discretization to the self-adjoint angular flux (SAAF) form of the multi-group neutron transport equation. The penalty parameters are determined, for general element types, from a mathematically rigorous coercivity analysis of the bilinear form. The proposed scheme produces a compact spatial discretization stencil. It also yields symmetric positive-definite (SPD) matrices, which can be efficiently solved using pre-conditioned conjugate gradient (PCG) solution algorithms. The proposed discretization scheme is verified using the method of manufactured solutions (MMS) and several nuclear reactor physics benchmark verification test cases. For sufficiently smooth elliptic problems, the proposed spatial discretization can exploit higher-order continuity, or k-refinement, of the NURBS basis to consistently yield greater numerical accuracy per degree of freedom (DoF) than standard h-refinement. Since this is a discontinuous scheme, it can also accurately model significant changes in the neutron scalar flux that may occur near the material interfaces of heterogeneous problems.

In this paper, our goal is to prove the existence of a weak solution (in ) for a fully nonlinear Dirichlet problem with a nonmonotone (e.g., Lipschitz) convection function F that depends on ∇ u , and a nonlinearity G that is not necessarily monotone and depends on the solution function u , and the higher order term is −ΔΓ( x , u ) − div a ( x , u , ∇ u ) provided that F , a , G , and Γ are Caratheòdory functions satisfying mild growth conditions. Here, Ω is a nonempty, bounded and open subset of R N with N ∈ Z + . We shall accomplish our goal by proving abstract existence results on the solvability of operator inclusion problems (in a Banach space X ) of the type and , where f ∈ X , is a β ‐expansive and m ‐accretive operator, is a L ‐Lipschitz continuous operator, is a compact operator and T : X ⟶ X is a β ‐expansive and continuous operator such that for some α &gt; 0, we have 〈 T u − T v , j ( u − v )〉 ≥ − α ‖ u − v ‖ 2 for all ( u , v ) ∈ X 2 and j ( u − v ) ∈ J ( u − v ). The proofs are mainly based on the recent result on surjectivity of compact perturbation of β ‐expansive operator due to Asfaw. The abstract results and applications are new and give improvements (and/or generalizations) of those known results.

In this paper, the standard multigroup neutron diffusion equations are derived as an asymptotic approximation to the multigroup neutron transport equations. The asymptotic analysis employs a scaling that (1) is suggested by the multigroup neutron diffusion equations themselves and (2) generalizes the long-known asymptotic scaling for monoenergetic transport problems. Two other asymptotic scalings of the multigroup transport equations are also considered, both of which lead to a new “group-collapsed” (monoenergetic) “equilibrium” diffusion approximation. The standard multigroup and equilibrium diffusion approximations are shown to preserve certain nonasymptotic properties of the multigroup transport equations. Generalizations of the analyses in this paper, and possible practical applications, are discussed.